To solve for the distance of the image from a concave mirror given the focal length and the object distance, we can use the mirror formula. The mirror formula is given by:
[tex]\[ \frac{1}{f} = \frac{1}{D_o} + \frac{1}{D_i} \][/tex]
Where:
- [tex]\( f \)[/tex] is the focal length of the mirror.
- [tex]\( D_o \)[/tex] is the object distance from the mirror.
- [tex]\( D_i \)[/tex] is the image distance from the mirror.
Given:
- The focal length [tex]\( f = 30 \)[/tex] cm.
- The object distance [tex]\( D_o = Y_o \)[/tex] cm.
We need to find the image distance [tex]\( D_i \)[/tex]. Let's rearrange the mirror formula to solve for [tex]\( D_i \)[/tex]:
[tex]\[ \frac{1}{D_i} = \frac{1}{f} - \frac{1}{D_o} \][/tex]
Now, substitute the given values into the formula:
[tex]\[ \frac{1}{D_i} = \frac{1}{30} - \frac{1}{Y_o} \][/tex]
To find [tex]\( D_i \)[/tex], take the reciprocal of the right-hand side:
[tex]\[ D_i = \frac{1}{\left( \frac{1}{30} - \frac{1}{Y_o} \right)} \][/tex]
This equation provides the distance of the image from the mirror based on the given focal length and the object distance [tex]\( Y_o \)[/tex].
To summarize, the distance of the image from the concave mirror [tex]\( D_i \)[/tex] can be calculated using the formula:
[tex]\[ D_i = \frac{1}{\left( \frac{1}{30} - \frac{1}{Y_o} \right)} \][/tex]
This is the final expression for [tex]\( D_i \)[/tex]. If you have a specific value for [tex]\( Y_o \)[/tex], you can substitute it into the formula to find the numerical value of [tex]\( D_i \)[/tex].
